Parallel Iterative Solvers with Localized ILU Preconditioning for Unstructured Grids on Workstation Clusters

Incomplete Lower-Upper (0) (ILU(O)) factorization is a very effective preconditioning method of iterative solvers for large-scale linear sparse systems in scientific and engineering computations. However, this method requires global data dependency, which is not ideal for parallel computations in which locality is of utmost importance.

In this paper, the localized ILU(O) preconditioning method is implemented in various types of iterative solvers. This method provides data locality and good parallelization on each processor. The performance of the developed system was evaluated on a workstation cluster using MPI.

In 1997, the Science and Technology Agency of Japan (STA) began a five-year project to develop the Earth Simulator. Both hardware and software for various types of global earth simulation are to be developed under this project. The present study was conducted as part of the research on software for solid earth simulation. This simulation code, named GeoFEM, solves crust deformation, mantle convection, seismic wave propagation, etc., using the finite element method.     

Direct numerical simulation of the amplification of a 2D temporal disturbance in plane Poiseuille flow

A direct numerical scheme is developed to study the temporal amplification of a 2D disturbance in plane Poiseuille flow. The transient non-linear Navier–Stokes equations are applied in a region of wavelength moving with the wave propagation speed. The complex amplitude involved in the perturbation functions is considered as the initial input of the non-linear stability equations. In this study a fully implicit finite difference scheme with five points in the flow direction and three points in the normal direction is developed so that numerical simulation of the amplification of a two-dimensional temporal disturbance in plane Poiseuille flow can be investigated. The growth and decay of the disturbance with time are presented and neutral stability curves which are in good agreement with existing solutions can be determined. The critical conditions as a function of the amplitude A₀ of the disturbance are presented. Fixing the wavelength, the Navier–Stokes equations are solved up to Re=10,000 a friction factor increasing with Reynolds number is observed. The 2D non-linear behaviour of the streamfunction, vorticity and velocity components at Re=10,000 are also exhibited. © 1998 John Wiley & Sons, Ltd.     

Equivalence Theorems on the Propagation of Small-Amplitude Waves in Prestressed Linearly Elastic Materials with Internal Constraints

By extending the procedure of linearization for constrained elastic materials in the papers by Marlow and Chadwick et al., we set up a linearized theory of constrained materials with initial stress (not necessarily based on a nonlinear theory). The conditions of propagation are characterized for small-displacement waves that may be either of discontinuity type of any given order or, in the homogeneous case, plane progressive. We see that, just as in the unconstrained case, the laws of propagation of discontinuity waves are the same as those of progressive waves. Waves are classified as mixed, kinematic, or ghost. Then we prove that the analogues of Truesdell"s two equivalence theorems on wave propagation in finite elasticity hold for each type of wave. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite element analysis of elastic-plastic plane stress problems based upon Tresca yield criterion

Summary A finite element technique for the elastic-plastic analysis of two dimensional structures subjected to conditions of plane stress and monotonically increasing loads is presented. The complete load deformation history as well as the propagation of the yield zones in the structure up to plastic collapse are studied. The material is assumed to be elastic-perfectly plastic and yielding is governed by Tresca yield condition. Plastic stress-strain relations for the sides and corners of Tresca yield condition are derived in terms of the components of the stresses and strains along a fixed reference coordinate system and the direction of the principal stress. The load is applied in small increments and the principal stress direction for each plastic element during a load increment is determined by an interpolation technique which leads to stresses that satisfy the yield condition. Numerical examples are given to illustrate the accuracy of the results obtained by the proposed method.

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Übersicht Es wird eine finite element-Methode vorgeschlagen zur Berechnung zweidimensionaler elastischplastischer Baukonstruktionen, die einem ebenen Spannungszustand mit monoton wachsender Last unterworfen sind. Das Belastungs-Verformungs-Verhalten und die Ausbreitung der Fließbereiche bis zum plastischen Zusammenbruch werden untersucht. Das Material soll der Fließbedingung von Tresca genügen. Spannungs-Verformungs-Berechnungen für die Ränder und Ecken des Tresca-Bereiches werden in Komponenten der Spannungen und Dehnungen längs der Achsriclitungen eines festen Bezugssystems und der Hauptspannungsrichtungen ausgedrückt. Die Belastung wird in kleinen Stufen aufgebracht, und die Hauptspannungsrichtungen werden für jedes plastische Teilchen während des Lastaufbringens durch Interpolation bestimmt. Die so erhaltenen Spannungen genügen der Fließbedingung. Durch numerische Beispiele wird die Genauigkeit des vorgeschlagenen Verfahrens demonstriert.

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On leave from Northwestern University, Evanston, Illinois     

Criteria of finite element algorithm for a class of parabolic equation

In finite element analysis of transient temperature field, it is quite notorious that the numerical solution may quite likely oscillate and/or exceed the reasonable scope, which violates the natural law of heat conduction. For this reason, we put forward the concept of time monotony and spatial monotony, and then derive several sufficient conditions for monotonic solutions in time dimension for 3 - D passive heat conduction equations with a group of finite difference schemes. For some special boundary conditions and regular element meshes, the lower and upper bounds for t/x ² can be obtained from those conditions so that reasonable numerical solutions are guaranteed. Spatial monotony is also discussed. Finally, the lumped mass method is analyzed. We creatively give several new criteria for the finite element solutions of a class of parabolic equation represented by heat conduction equation.     

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations

Superposition of non-ordinary state-based peridynamics and finite element method for material failure simulations, including crack propagation and strain localization is developed. By this approach, a peridynamic model capable of effectively treating strong and weak discontinuities is superimposed in the critical regions over an underlying finite element mesh placed over the entire problem domain. A rigorous variational framework of coupling local finite element and nonlocal peridynamics approximations that is free of blending parameters is developed. Several numerical examples involving mixed-model fracture, three-dimensional adaptive crack propagation and strain localization induced ductile failure demonstrate the rational and efficiency of the proposed superposition-based coupling approach.

     

黏弹性人工边界等效荷载计算的改进方法

黏弹性人工边界在场地地震反应和结构-地基动力相互作用等问题的计算中已得到了广泛的应用.地震波在黏弹性人工边界中的输入是通过将地震波转化为作用于人工边界处的等效载荷来实现的.计算等效节点载荷的常规方法默认边界节点对应区域内的应力为均布力,但实际上该节点对应区域内的应力分布通常是不均匀的.本文在有限元方法结合黏弹性局部人工边界的显式时域波动方法的基础上,建立了无限域散射问题地震波等效载荷计算的一种改进方法.该方法采用细化网格与应力积分相结合的方法计算人工边界等效节点力,有效地降低了人工边界上等效节点力的计算误差.以不同角度入射地震波的二维算例为例,算例给出的波场位移云图和节点位移时程曲线验证了本文方法的有效性,其计算精度与网格尺寸和地震波入射角度密切相关,且网格越小、入射角度越小,计算精度越高.对于相同的网格尺寸,本文采用方法的计算精度明显高于常规方法,尤其是对于斜入射问题优势更为明显.     

Contribution to the dynamics of a horizontally unstretchable half-space of the westergaard type

Summary The transfer matrix of a layer is derived from the dynamic equations of a continuum reinforced with very thin unstretchable membranes. For the case of a periodically spaced reinforcement, the finite difference method enables to find closed solutions. The elastic wave propagation in a half-space is investigated, and it is shown that the only possible nonattenuating wave has the velocityc ₂ of the shear wave.     

On a magneto-elastic system with discontinuous coefficients and the propagation of a weak discontinuity

Summary In this paper we deduce the magneto-elastic system of equations in the three-dimensional case. As an application we study the propagation of a weak discontinuity when a strong discontinuity also occurs.

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Sommario In questa Nota si deduce un sistema di equazioni per la Magneto-elasticità con deformazioni finite nel caso tridimensionale. Come Applicazione si studia la propagazione di una discontinuità debole nella ipotesi che i coefficienti delle equazioni risultano essi stessi discontinui. Tale ipotesi risulta verificata, ad es., quando si considerano due differenti mezzi magneto-elastici uno in contatto con l'altro.

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Work supported by the CNR through the Gruppo Nazionale per la Fisica-Matematica.     

Evaluation of crack initiation angle under mixed mode loading at diverse strain rates

Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k₁ and k₂. Results are in agreement with those observed experimentally.     

Resolution of linear viscoelastic equations in the frequency domain using real Helmholtz boundary integral equations

Boundary integral equations are well suitable for the analysis of seismic waves propagation in unbounded domains. Formulations in elastodynamics are well developed. In contrast, for the dynamic analysis of viscoelastic media, there are very seldom formulations by boundary integral equations. In this Note, we propose a new and simple formulation of time harmonic viscoelasticity with the Zener model, which reduces to classical elastodynamics if a compatibility condition is satisfied by boundary conditions. Intermediate variables which satisfy the classical elastodynamic equations are introduced. It makes it possible to utilize existing numerical tools of time harmonic elastodynamics. To cite this article: S. Chaillat, H.D. Bui, C. R. Mecanique 335 (2007).     

ANCF curvature continuity: application to soft and fluid materials

The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shown that continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained using the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials.

     

The effect of a solid wall for the flow of dilute macromolecular solutions

Summary In the vicinity of a wall the behavior of the molecules of a fluid is not only influenced by neighboring molecules but also by the wall. For a macromolecular solution this implies that the constitutive equations cannot be employed in any finite system except for the calculation of average quantities if, in addition, one accounts for a velocity slip at the solid wall. For planeCouette flow of a dilute solution of flexible macromolecules idealized asHookean dumbbells the slip velocity,u _s , is evaluated under the assumption that the interaction between the solute molecules and the wall is a hard core repulsion. This furnishes a slip velocity in the direction of the flow so that the apparent viscosity goes down by increasing the ratio of the length of the molecules to the distance between the plates. Increasing the intrinsic viscosity and/or the concentration will enhance that effect.

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Zusammenfassung In der Nähe einer Wand wird das Verhalten der Moleküle einer Flüssigkeit nicht nur von Nachbarmolekülen, sondern auch von der Wand beeinflußt. Für eine makromolekulare Lösung bedeutet das, daß die rheologischen Stoffgleichungen in einem endlichen System nicht anwendbar sind. Zur Berechnung gemittelter Größen kann sie jedoch noch verwendet werden, vorausgesetzt, man zieht eine Geschwindigkeitsgleitung mit in die Betrachtung ein. Für die ebeneCouette-Strömung einer verdünnten Lösung flexibler Makromoleküle, die man durchHookesche Hanteln idealisiert, läßt sich die Gleitungsgeschwindigkeitu _s berechnen unter der Annahme, daß die Wechselwirkung mit der Wand rein abstoßend ist. Es zeigt sich, daß dies eine Gleitungsgeschwindigkeit in Strömungsrichtung nach sich zieht, so daß sich die scheinbare Viskosität der Strömung bei größer werdendem Verhältnis von Länge der Moleküle zum Plattenabstand erniedrigt. Eine Vergrößerung desStaudinger-Index und/oder der Konzentration vergrößert den Effekt.

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Paper presented at the Annual Conference of German Rheologists in Berlin, April 28–30, 1975.

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With 2 figures     

A comparative study of two fast nonlinear free‐surface water wave models

This paper presents a comparison in terms of accuracy and efficiency between two fully nonlinear potential flow solvers for the solution of gravity wave propagation. One model is based on the high‐order spectral (HOS) method, whereas the second model is the high‐order finite difference model OceanWave3D. Although both models solve the nonlinear potential flow problem, they make use of two different approaches. The HOS model uses a modal expansion in the vertical direction to collapse the numerical solution to the two‐dimensional horizontal plane. On the other hand, the finite difference model simply directly solves the three‐dimensional problem. Both models have been well validated on standard test cases and shown to exhibit attractive convergence properties and an optimal scaling of the computational effort with increasing problem size. These two models are compared for solution of a typical problem: propagation of highly nonlinear periodic waves on a finite constant‐depth domain. The HOS model is found to be more efficient than OceanWave3D with a difference dependent on the level of accuracy needed as well as the wave steepness. Also, the higher the order of the finite difference schemes used in OceanWave3D, the closer the results come to the HOS model. Copyright © 2011 John Wiley & Sons, Ltd.     

Mathematical Justification of the Obstacle Problem in the Case of a Shallow Shell

This paper deals with the asymptotic formulation and justification of a mechanical model for a shallow shell in frictionless unilateral contact with an obstacle. The first three parts of the paper concern the formulation of the equilibrium problem. Special attention is paid to the contact conditions, which are usual within two or three dimensional elasticity, but which are not so usual in shell theories. Lastly the limit problem is formulated in the main part of the paper and a convergence result is presented. Two points are worth stressing here. First, we point out that unlike classical bilateral shell models justifications, the functional framework of the present analysis involves cones. Secondly, while the cones result from a positivity condition on the boundary as long as the thickness parameter is finite, leading to a Signorini problem in the Sobolev space H ¹, the cone results from a positivity condition in the domain, giving rise to a so-called obstacle problem in the Sobolev space H ² at the limit.      

Design of a tuned vibration absorber for a slender hollow cylindrical structure

Abstract

In this paper, details of the design work for a tuned vibration absorber to be used on a hollow cylindrical structure is presented. The vibration problem is of resonant type and the tuned vibration absorber is designed to suppress the displacement vibration response of the free end of the slender hollow structure dominated by the contribution of its lowest transverse vibration modes. The structure is modeled using a commercial finite element software. Finite element model of the structure is verified using experimentally obtained frequency response functions and modal parameters. Effective parameters of the tuned vibration absorber design are then determined based on finite element analysis simulations of the vibration suppression performance of the tuned vibration absorber as it is used on the structure. Details of the tuned vibration absorber design are determined and a prototype is fabricated. Prototype tuned vibration absorber is then characterized experimentally both as a standalone system and also as it is used on the main structure. Vibration reduction performance of the physical prototype of the tuned vibration absorber is also compared with its vibration reduction performance estimated from finite element analysis simulations so that the analysis based design process can be validated.

Communicated by Dumitru Caruntu.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

Numerical investigation of a third-order ODE from thin film flow

We compare two finite difference schemes to solve the third-order ordinary differential equation y'=y ^−k from thin film flow. The boundary conditions come from Tanner’s problem for the surface tension driven flow of a thin film. We show that a central difference approximation to the third derivative in the model equation produces a solution curve with oscillations. A difference scheme based on a combination of forward and backward differences produces a smooth accurate solution curve. Both the 0-stability and von Neumann stability properties of the different finite difference schemes are analyzed. The solution curves obtained from both approaches are presented and discussed.     

The crack resistance of a dilatation-sensitive material

The entire history of crack propagation in high-strength steel with dilatation-sensitivity is investigated in this work. Based on the experimentally determined stress-strain curves varying in the strain history according to the mean stress level, a central cracked panel made of unaged maraging steel subject to a remote tension is considered in the analysis. Due to tremendous amount of hydrostatic stress established in the vicinity of the crack tip, the effect of mean stress is accounted for in the onset for plastic flow. The J₂−I₁ non-associate flow rule is incorporated in the incremental analysis employing the isoparametric finite element method with displacement formulation. When the stage of global instability is approached, the possible formation of a secondary crack in front of the primary crack is demonstrated by the use of strain energy density criterion. Also, the size effect on the crack resistance curve is established at the end for the unaged maraging steel.